LMFDB - Embedded newform 2106.2.b.c.649.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2106,2,Mod(649,2106)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2106.649"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2106, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2106 = 2 \cdot 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2106.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [14,0,0,-14,0,0,0,0,0,0,0,0,-2,-8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(14)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$16.8164946657$
Analytic rank:	$0$
Dimension:	$14$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 34x^{12} + 435x^{10} + 2617x^{8} + 7651x^{6} + 10260x^{4} + 5589x^{2} + 729 $ x^14 + 34x^12 + 435x^10 + 2617x^8 + 7651x^6 + 10260x^4 + 5589x^2 + 729
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 234)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.1
Root			$2.42952i$ of defining polynomial
Character	$\chi$	$=$	2106.649
Dual form			2106.2.b.c.649.14

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.00000i q^{2} -1.00000 q^{4} -3.15852i q^{5} +1.57492i q^{7} +1.00000i q^{8} -3.15852 q^{10} -3.76656i q^{11} +(2.57405 + 2.52473i) q^{13} +1.57492 q^{14} +1.00000 q^{16} +7.06565 q^{17} -3.76656i q^{19} +3.15852i q^{20} -3.76656 q^{22} +3.69746 q^{23} -4.97624 q^{25} +(2.52473 - 2.57405i) q^{26} -1.57492i q^{28} -0.218255 q^{29} +3.06910i q^{31} -1.00000i q^{32} -7.06565i q^{34} +4.97440 q^{35} -0.292126i q^{37} -3.76656 q^{38} +3.15852 q^{40} -7.38168i q^{41} +6.11671 q^{43} +3.76656i q^{44} -3.69746i q^{46} -7.13476i q^{47} +4.51964 q^{49} +4.97624i q^{50} +(-2.57405 - 2.52473i) q^{52} -14.4175 q^{53} -11.8968 q^{55} -1.57492 q^{56} +0.218255i q^{58} +10.4471i q^{59} -6.00013 q^{61} +3.06910 q^{62} -1.00000 q^{64} +(7.97440 - 8.13019i) q^{65} -7.31599i q^{67} -7.06565 q^{68} -4.97440i q^{70} -0.772410i q^{71} -13.5342i q^{73} -0.292126 q^{74} +3.76656i q^{76} +5.93202 q^{77} -12.6807 q^{79} -3.15852i q^{80} -7.38168 q^{82} +0.363495i q^{83} -22.3170i q^{85} -6.11671i q^{86} +3.76656 q^{88} +7.06555i q^{89} +(-3.97624 + 4.05392i) q^{91} -3.69746 q^{92} -7.13476 q^{94} -11.8968 q^{95} +0.618592i q^{97} -4.51964i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64}+ \cdots - 8 q^{92}+O(q^{100}) $ 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2106\mathbb{Z}\right)^\times$.

$n$	$1379$	$1783$
$\chi(n)$	$1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		−	1.00000i		−	0.707107i
$2$		−	1.00000i		−	0.707107i
$3$		0			0
$3$		0			0
$4$	−1.00000			−0.500000
$5$		−	3.15852i		−	1.41253i	−0.707946	−	0.706266i	$-0.750378\pi$
$5$		−	3.15852i		−	1.41253i	0.707946	−	0.706266i	$-0.249622\pi$
$6$		0			0
$7$			1.57492i			0.595262i	0.954681	+	0.297631i	$0.0961966\pi$
$7$			1.57492i			0.595262i	−0.954681	+	0.297631i	$0.903803\pi$
$8$			1.00000i			0.353553i
$9$		0			0
$10$	−3.15852			−0.998811
$11$		−	3.76656i		−	1.13566i	−0.823146	−	0.567830i	$-0.807783\pi$
$11$		−	3.76656i		−	1.13566i	0.823146	−	0.567830i	$-0.192217\pi$
$12$		0			0
$13$	2.57405	+	2.52473i	0.713914	+	0.700234i
$13$	2.57405	+	2.52473i	0.713914	+	0.700234i
$14$	1.57492			0.420914
$15$		0			0
$16$	1.00000			0.250000
$17$	7.06565			1.71367			0.856836	−	0.515588i	$-0.172427\pi$
$17$	7.06565			1.71367			0.856836	+	0.515588i	$0.172427\pi$
$18$		0			0
$19$		−	3.76656i		−	0.864108i	−0.901848	−	0.432054i	$-0.857789\pi$
$19$		−	3.76656i		−	0.864108i	0.901848	−	0.432054i	$-0.142211\pi$
$20$			3.15852i			0.706266i
$21$		0			0
$22$	−3.76656			−0.803033
$23$	3.69746			0.770974			0.385487	−	0.922713i	$-0.374033\pi$
$23$	3.69746			0.770974			0.385487	+	0.922713i	$0.374033\pi$
$24$		0			0
$25$	−4.97624			−0.995247
$26$	2.52473	−	2.57405i	0.495140	−	0.504813i
$27$		0			0
$28$		−	1.57492i		−	0.297631i
$29$	−0.218255			−0.0405290			−0.0202645	−	0.999795i	$-0.506451\pi$
$29$	−0.218255			−0.0405290			−0.0202645	+	0.999795i	$0.506451\pi$
$30$		0			0
$31$			3.06910i			0.551227i	0.961269	+	0.275613i	$0.0888809\pi$
$31$			3.06910i			0.551227i	−0.961269	+	0.275613i	$0.911119\pi$
$32$		−	1.00000i		−	0.176777i
$33$		0			0
$34$		−	7.06565i		−	1.21175i
$35$	4.97440			0.840827
$36$		0			0
$37$		−	0.292126i		−	0.0480251i	−0.999712	−	0.0240126i	$-0.992356\pi$
$37$		−	0.292126i		−	0.0480251i	0.999712	−	0.0240126i	$-0.00764417\pi$
$38$	−3.76656			−0.611017
$39$		0			0
$40$	3.15852			0.499406
$41$		−	7.38168i		−	1.15282i	−0.817159	−	0.576412i	$-0.804452\pi$
$41$		−	7.38168i		−	1.15282i	0.817159	−	0.576412i	$-0.195548\pi$
$42$		0			0
$43$	6.11671			0.932789			0.466395	−	0.884577i	$-0.345553\pi$
$43$	6.11671			0.932789			0.466395	+	0.884577i	$0.345553\pi$
$44$			3.76656i			0.567830i
$45$		0			0
$46$		−	3.69746i		−	0.545161i
$47$		−	7.13476i		−	1.04071i	−0.853950	−	0.520356i	$-0.825799\pi$
$47$		−	7.13476i		−	1.04071i	0.853950	−	0.520356i	$-0.174201\pi$
$48$		0			0
$49$	4.51964			0.645663
$50$			4.97624i			0.703746i
$51$		0			0
$52$	−2.57405	−	2.52473i	−0.356957	−	0.350117i
$53$	−14.4175			−1.98040			−0.990201	−	0.139649i	$-0.955403\pi$
$53$	−14.4175			−1.98040			−0.990201	+	0.139649i	$0.955403\pi$
$54$		0			0
$55$	−11.8968			−1.60416
$56$	−1.57492			−0.210457
$57$		0			0
$58$			0.218255i			0.0286583i
$59$			10.4471i			1.36010i	0.733168	+	0.680048i	$0.238041\pi$
$59$			10.4471i			1.36010i	−0.733168	+	0.680048i	$0.761959\pi$
$60$		0			0
$61$	−6.00013			−0.768239			−0.384119	−	0.923283i	$-0.625495\pi$
$61$	−6.00013			−0.768239			−0.384119	+	0.923283i	$0.625495\pi$
$62$	3.06910			0.389776
$63$		0			0
$64$	−1.00000			−0.125000
$65$	7.97440	−	8.13019i	0.989103	−	1.00843i
$66$		0			0
$67$		−	7.31599i		−	0.893790i	−0.894586	−	0.446895i	$-0.852530\pi$
$67$		−	7.31599i		−	0.893790i	0.894586	−	0.446895i	$-0.147470\pi$
$68$	−7.06565			−0.856836
$69$		0			0
$70$		−	4.97440i		−	0.594555i
$71$		−	0.772410i		−	0.0916682i	−0.998949	−	0.0458341i	$-0.985405\pi$
$71$		−	0.772410i		−	0.0916682i	0.998949	−	0.0458341i	$-0.0145946\pi$
$72$		0			0
$73$		−	13.5342i		−	1.58406i	−0.610480	−	0.792032i	$-0.709024\pi$
$73$		−	13.5342i		−	1.58406i	0.610480	−	0.792032i	$-0.290976\pi$
$74$	−0.292126			−0.0339589
$75$		0			0
$76$			3.76656i			0.432054i
$77$	5.93202			0.676016
$78$		0			0
$79$	−12.6807			−1.42669			−0.713343	−	0.700815i	$-0.752820\pi$
$79$	−12.6807			−1.42669			−0.713343	+	0.700815i	$0.752820\pi$
$80$		−	3.15852i		−	0.353133i
$81$		0			0
$82$	−7.38168			−0.815170
$83$			0.363495i			0.0398987i	0.999801	+	0.0199494i	$0.00635050\pi$
$83$			0.363495i			0.0398987i	−0.999801	+	0.0199494i	$0.993650\pi$
$84$		0			0
$85$		−	22.3170i		−	2.42062i
$86$		−	6.11671i		−	0.659581i
$87$		0			0
$88$	3.76656			0.401517
$89$			7.06555i			0.748947i	0.927238	+	0.374473i	$0.122176\pi$
$89$			7.06555i			0.748947i	−0.927238	+	0.374473i	$0.877824\pi$
$90$		0			0
$91$	−3.97624	+	4.05392i	−0.416823	+	0.424966i
$92$	−3.69746			−0.385487
$93$		0			0
$94$	−7.13476			−0.735894
$95$	−11.8968			−1.22058
$96$		0			0
$97$			0.618592i			0.0628085i	0.999507	+	0.0314042i	$0.00999792\pi$
$97$			0.618592i			0.0628085i	−0.999507	+	0.0314042i	$0.990002\pi$
$98$		−	4.51964i		−	0.456552i
$99$		0			0
$100$	4.97624			0.497624
$101$	8.33563			0.829426			0.414713	−	0.909952i	$-0.363882\pi$
$101$	8.33563			0.829426			0.414713	+	0.909952i	$0.363882\pi$
$102$		0			0
$103$	−9.89675			−0.975156			−0.487578	−	0.873079i	$-0.662119\pi$
$103$	−9.89675			−0.975156			−0.487578	+	0.873079i	$0.662119\pi$
$104$	−2.52473	+	2.57405i	−0.247570	+	0.252407i
$105$		0			0
$106$			14.4175i			1.40036i
$107$	0.115559			0.0111715			0.00558574	−	0.999984i	$-0.498222\pi$
$107$	0.115559			0.0111715			0.00558574	+	0.999984i	$0.498222\pi$
$108$		0			0
$109$			14.1301i			1.35342i	0.736251	+	0.676708i	$0.236594\pi$
$109$			14.1301i			1.35342i	−0.736251	+	0.676708i	$0.763406\pi$
$110$			11.8968i			1.13431i
$111$		0			0
$112$			1.57492i			0.148816i
$113$	19.4194			1.82682			0.913410	−	0.407041i	$-0.133439\pi$
$113$	19.4194			1.82682			0.913410	+	0.407041i	$0.133439\pi$
$114$		0			0
$115$		−	11.6785i		−	1.08903i
$116$	0.218255			0.0202645
$117$		0			0
$118$	10.4471			0.961733
$119$			11.1278i			1.02009i
$120$		0			0
$121$	−3.18698			−0.289726
$122$			6.00013i			0.543227i
$123$		0			0
$124$		−	3.06910i		−	0.275613i
$125$		−	0.0750557i		−	0.00671318i
$126$		0			0
$127$	5.42904			0.481750			0.240875	−	0.970556i	$-0.422566\pi$
$127$	5.42904			0.481750			0.240875	+	0.970556i	$0.422566\pi$
$128$			1.00000i			0.0883883i
$129$		0			0
$130$	−8.13019	−	7.97440i	−0.713065	−	0.699401i
$131$	−12.6868			−1.10845			−0.554227	−	0.832365i	$-0.686986\pi$
$131$	−12.6868			−1.10845			−0.554227	+	0.832365i	$0.686986\pi$
$132$		0			0
$133$	5.93202			0.514371
$134$	−7.31599			−0.632005
$135$		0			0
$136$			7.06565i			0.605875i
$137$			7.35541i			0.628415i	0.949354	+	0.314207i	$0.101739\pi$
$137$			7.35541i			0.628415i	−0.949354	+	0.314207i	$0.898261\pi$
$138$		0			0
$139$	−7.31119			−0.620127			−0.310063	−	0.950716i	$-0.600350\pi$
$139$	−7.31119			−0.620127			−0.310063	+	0.950716i	$0.600350\pi$
$140$	−4.97440			−0.420414
$141$		0			0
$142$	−0.772410			−0.0648192
$143$	9.50955	−	9.69532i	0.795228	−	0.810764i
$144$		0			0
$145$			0.689364i			0.0572485i
$146$	−13.5342			−1.12010
$147$		0			0
$148$			0.292126i			0.0240126i
$149$		−	23.7638i		−	1.94680i	−0.229107	−	0.973401i	$-0.573580\pi$
$149$		−	23.7638i		−	1.94680i	0.229107	−	0.973401i	$-0.426420\pi$
$150$		0			0
$151$		−	10.3890i		−	0.845442i	−0.906260	−	0.422721i	$-0.861075\pi$
$151$		−	10.3890i		−	0.845442i	0.906260	−	0.422721i	$-0.138925\pi$
$152$	3.76656			0.305508
$153$		0			0
$154$		−	5.93202i		−	0.478016i
$155$	9.69381			0.778626
$156$		0			0
$157$	−9.14824			−0.730109			−0.365054	−	0.930986i	$-0.618950\pi$
$157$	−9.14824			−0.730109			−0.365054	+	0.930986i	$0.618950\pi$
$158$			12.6807i			1.00882i
$159$		0			0
$160$	−3.15852			−0.249703
$161$			5.82319i			0.458932i
$162$		0			0
$163$		−	13.5935i		−	1.06473i	−0.846516	−	0.532363i	$-0.821304\pi$
$163$		−	13.5935i		−	1.06473i	0.846516	−	0.532363i	$-0.178696\pi$
$164$			7.38168i			0.576412i
$165$		0			0
$166$	0.363495			0.0282126
$167$			7.16615i			0.554534i	0.960793	+	0.277267i	$0.0894286\pi$
$167$			7.16615i			0.554534i	−0.960793	+	0.277267i	$0.910571\pi$
$168$		0			0
$169$	0.251488	+	12.9976i	0.0193452	+	0.999813i
$170$	−22.3170			−1.71164
$171$		0			0
$172$	−6.11671			−0.466395
$173$	3.01163			0.228970			0.114485	−	0.993425i	$-0.463478\pi$
$173$	3.01163			0.228970			0.114485	+	0.993425i	$0.463478\pi$
$174$		0			0
$175$		−	7.83716i		−	0.592433i
$176$		−	3.76656i		−	0.283915i
$177$		0			0
$178$	7.06555			0.529585
$179$	−19.2107			−1.43587			−0.717936	−	0.696109i	$-0.754913\pi$
$179$	−19.2107			−1.43587			−0.717936	+	0.696109i	$0.754913\pi$
$180$		0			0
$181$	9.14810			0.679973			0.339987	−	0.940430i	$-0.389577\pi$
$181$	9.14810			0.679973			0.339987	+	0.940430i	$0.389577\pi$
$182$	4.05392	+	3.97624i	0.300496	+	0.294738i
$183$		0			0
$184$			3.69746i			0.272580i
$185$	−0.922684			−0.0678371
$186$		0			0
$187$		−	26.6132i		−	1.94615i
$188$			7.13476i			0.520356i
$189$		0			0
$190$			11.8968i			0.863081i
$191$	0.263254			0.0190484			0.00952420	−	0.999955i	$-0.496968\pi$
$191$	0.263254			0.0190484			0.00952420	+	0.999955i	$0.496968\pi$
$192$		0			0
$193$			11.7167i			0.843390i	0.906738	+	0.421695i	$0.138565\pi$
$193$			11.7167i			0.843390i	−0.906738	+	0.421695i	$0.861435\pi$
$194$	0.618592			0.0444123
$195$		0			0
$196$	−4.51964			−0.322831
$197$		−	20.5283i		−	1.46258i	−0.682066	−	0.731291i	$-0.738918\pi$
$197$		−	20.5283i		−	1.46258i	0.682066	−	0.731291i	$-0.261082\pi$
$198$		0			0
$199$	5.95247			0.421960			0.210980	−	0.977490i	$-0.432335\pi$
$199$	5.95247			0.421960			0.210980	+	0.977490i	$0.432335\pi$
$200$		−	4.97624i		−	0.351873i
$201$		0			0
$202$		−	8.33563i		−	0.586493i
$203$		−	0.343734i		−	0.0241254i
$204$		0			0
$205$	−23.3152			−1.62840
$206$			9.89675i			0.689539i
$207$		0			0
$208$	2.57405	+	2.52473i	0.178478	+	0.175058i
$209$	−14.1870			−0.981334
$210$		0			0
$211$	−17.3925			−1.19735			−0.598674	−	0.800993i	$-0.704306\pi$
$211$	−17.3925			−1.19735			−0.598674	+	0.800993i	$0.704306\pi$
$212$	14.4175			0.990201
$213$		0			0
$214$		−	0.115559i		−	0.00789943i
$215$		−	19.3197i		−	1.31759i
$216$		0			0
$217$	−4.83358			−0.328125
$218$	14.1301			0.957010
$219$		0			0
$220$	11.8968			0.802079
$221$	18.1874	+	17.8389i	1.22341	+	1.19997i
$222$		0			0
$223$			23.7456i			1.59012i	0.606528	+	0.795062i	$0.292562\pi$
$223$			23.7456i			1.59012i	−0.606528	+	0.795062i	$0.707438\pi$
$224$	1.57492			0.105229
$225$		0			0
$226$		−	19.4194i		−	1.29176i
$227$			4.54135i			0.301420i	0.988578	+	0.150710i	$0.0481559\pi$
$227$			4.54135i			0.301420i	−0.988578	+	0.150710i	$0.951844\pi$
$228$		0			0
$229$		−	11.9711i		−	0.791074i	−0.918450	−	0.395537i	$-0.870558\pi$
$229$		−	11.9711i		−	0.791074i	0.918450	−	0.395537i	$-0.129442\pi$
$230$	−11.6785			−0.770057
$231$		0			0
$232$		−	0.218255i		−	0.0143292i
$233$	10.6895			0.700290			0.350145	−	0.936696i	$-0.386132\pi$
$233$	10.6895			0.700290			0.350145	+	0.936696i	$0.386132\pi$
$234$		0			0
$235$	−22.5353			−1.47004
$236$		−	10.4471i		−	0.680048i
$237$		0			0
$238$	11.1278			0.721309
$239$		−	24.0968i		−	1.55869i	−0.626594	−	0.779346i	$-0.715552\pi$
$239$		−	24.0968i		−	1.55869i	0.626594	−	0.779346i	$-0.284448\pi$
$240$		0			0
$241$			11.1299i			0.716941i	0.933541	+	0.358470i	$0.116702\pi$
$241$			11.1299i			0.716941i	−0.933541	+	0.358470i	$0.883298\pi$
$242$			3.18698i			0.204867i
$243$		0			0
$244$	6.00013			0.384119
$245$		−	14.2754i		−	0.912019i
$246$		0			0
$247$	9.50955	−	9.69532i	0.605078	−	0.616899i
$248$	−3.06910			−0.194888
$249$		0			0
$250$	−0.0750557			−0.00474694
$251$	11.3356			0.715499			0.357749	−	0.933818i	$-0.383544\pi$
$251$	11.3356			0.715499			0.357749	+	0.933818i	$0.383544\pi$
$252$		0			0
$253$		−	13.9267i		−	0.875565i
$254$		−	5.42904i		−	0.340648i
$255$		0			0
$256$	1.00000			0.0625000
$257$	8.32053			0.519020			0.259510	−	0.965740i	$-0.416439\pi$
$257$	8.32053			0.519020			0.259510	+	0.965740i	$0.416439\pi$
$258$		0			0
$259$	0.460073			0.0285876
$260$	−7.97440	+	8.13019i	−0.494551	+	0.504213i
$261$		0			0
$262$			12.6868i			0.783796i
$263$	−14.3827			−0.886872			−0.443436	−	0.896306i	$-0.646241\pi$
$263$	−14.3827			−0.886872			−0.443436	+	0.896306i	$0.646241\pi$
$264$		0			0
$265$			45.5381i			2.79738i
$266$		−	5.93202i		−	0.363715i
$267$		0			0
$268$			7.31599i			0.446895i
$269$	9.01366			0.549572			0.274786	−	0.961505i	$-0.411393\pi$
$269$	9.01366			0.549572			0.274786	+	0.961505i	$0.411393\pi$
$270$		0			0
$271$			19.9094i			1.20941i	0.796449	+	0.604705i	$0.206709\pi$
$271$			19.9094i			1.20941i	−0.796449	+	0.604705i	$0.793291\pi$
$272$	7.06565			0.428418
$273$		0			0
$274$	7.35541			0.444356
$275$			18.7433i			1.13026i
$276$		0			0
$277$	13.3831			0.804115			0.402058	−	0.915614i	$-0.368295\pi$
$277$	13.3831			0.804115			0.402058	+	0.915614i	$0.368295\pi$
$278$			7.31119i			0.438496i
$279$		0			0
$280$			4.97440i			0.297277i
$281$			0.700291i			0.0417759i	0.999782	+	0.0208879i	$0.00664932\pi$
$281$			0.700291i			0.0417759i	−0.999782	+	0.0208879i	$0.993351\pi$
$282$		0			0
$283$	19.4357			1.15533			0.577665	−	0.816274i	$-0.303964\pi$
$283$	19.4357			1.15533			0.577665	+	0.816274i	$0.303964\pi$
$284$			0.772410i			0.0458341i
$285$		0			0
$286$	−9.69532	−	9.50955i	−0.573297	−	0.562311i
$287$	11.6255			0.686233
$288$		0			0
$289$	32.9235			1.93668
$290$	0.689364			0.0404808
$291$		0			0
$292$			13.5342i			0.792032i
$293$			4.25760i			0.248732i	0.992236	+	0.124366i	$0.0396897\pi$
$293$			4.25760i			0.248732i	−0.992236	+	0.124366i	$0.960310\pi$
$294$		0			0
$295$	32.9973			1.92118
$296$	0.292126			0.0169795
$297$		0			0
$298$	−23.7638			−1.37660
$299$	9.51746	+	9.33509i	0.550409	+	0.539862i
$300$		0			0
$301$			9.63330i			0.555254i
$302$	−10.3890			−0.597818
$303$		0			0
$304$		−	3.76656i		−	0.216027i
$305$			18.9515i			1.08516i
$306$		0			0
$307$		−	7.35088i		−	0.419537i	−0.977751	−	0.209768i	$-0.932729\pi$
$307$		−	7.35088i		−	0.419537i	0.977751	−	0.209768i	$-0.0672710\pi$
$308$	−5.93202			−0.338008
$309$		0			0
$310$		−	9.69381i		−	0.550571i
$311$	−2.81511			−0.159630			−0.0798152	−	0.996810i	$-0.525433\pi$
$311$	−2.81511			−0.159630			−0.0798152	+	0.996810i	$0.525433\pi$
$312$		0			0
$313$	27.0917			1.53131			0.765657	−	0.643249i	$-0.222414\pi$
$313$	27.0917			1.53131			0.765657	+	0.643249i	$0.222414\pi$
$314$			9.14824i			0.516265i
$315$		0			0
$316$	12.6807			0.713343
$317$			0.756736i			0.0425025i	0.999774	+	0.0212513i	$0.00676499\pi$
$317$			0.756736i			0.0425025i	−0.999774	+	0.0212513i	$0.993235\pi$
$318$		0			0
$319$			0.822072i			0.0460272i
$320$			3.15852i			0.176567i
$321$		0			0
$322$	5.82319			0.324514
$323$		−	26.6132i		−	1.48080i
$324$		0			0
$325$	−12.8091	−	12.5636i	−0.710521	−	0.696906i
$326$	−13.5935			−0.752875
$327$		0			0
$328$	7.38168			0.407585
$329$	11.2366			0.619496
$330$		0			0
$331$		−	32.9672i		−	1.81204i	−0.423234	−	0.906021i	$-0.639105\pi$
$331$		−	32.9672i		−	1.81204i	0.423234	−	0.906021i	$-0.360895\pi$
$332$		−	0.363495i		−	0.0199494i
$333$		0			0
$334$	7.16615			0.392114
$335$	−23.1077			−1.26251
$336$		0			0
$337$	4.50695			0.245509			0.122755	−	0.992437i	$-0.460827\pi$
$337$	4.50695			0.245509			0.122755	+	0.992437i	$0.460827\pi$
$338$	12.9976	−	0.251488i	0.706974	−	0.0136791i
$339$		0			0
$340$			22.3170i			1.21031i
$341$	11.5600			0.626007
$342$		0			0
$343$			18.1425i			0.979601i
$344$			6.11671i			0.329791i
$345$		0			0
$346$		−	3.01163i		−	0.161906i
$347$	−29.9717			−1.60897			−0.804484	−	0.593975i	$-0.797558\pi$
$347$	−29.9717			−1.60897			−0.804484	+	0.593975i	$0.797558\pi$
$348$		0			0
$349$			17.4351i			0.933278i	0.884448	+	0.466639i	$0.154535\pi$
$349$			17.4351i			0.933278i	−0.884448	+	0.466639i	$0.845465\pi$
$350$	−7.83716			−0.418914
$351$		0			0
$352$	−3.76656			−0.200758
$353$			4.51846i			0.240493i	0.992744	+	0.120247i	$0.0383685\pi$
$353$			4.51846i			0.240493i	−0.992744	+	0.120247i	$0.961631\pi$
$354$		0			0
$355$	−2.43967			−0.129484
$356$		−	7.06555i		−	0.374473i
$357$		0			0
$358$			19.2107i			1.01531i
$359$			30.5681i			1.61332i	0.591013	+	0.806662i	$0.298728\pi$
$359$			30.5681i			1.61332i	−0.591013	+	0.806662i	$0.701272\pi$
$360$		0			0
$361$	4.81302			0.253317
$362$		−	9.14810i		−	0.480814i
$363$		0			0
$364$	3.97624	−	4.05392i	0.208411	−	0.212483i
$365$	−42.7481			−2.23754
$366$		0			0
$367$	−10.6684			−0.556886			−0.278443	−	0.960453i	$-0.589818\pi$
$367$	−10.6684			−0.556886			−0.278443	+	0.960453i	$0.589818\pi$
$368$	3.69746			0.192743
$369$		0			0
$370$			0.922684i			0.0479681i
$371$		−	22.7064i		−	1.17886i
$372$		0			0
$373$	8.36446			0.433095			0.216548	−	0.976272i	$-0.430520\pi$
$373$	8.36446			0.433095			0.216548	+	0.976272i	$0.430520\pi$
$374$	−26.6132			−1.37614
$375$		0			0
$376$	7.13476			0.367947
$377$	−0.561801	−	0.551036i	−0.0289342	−	0.0283798i
$378$		0			0
$379$			26.5494i			1.36375i	0.731467	+	0.681877i	$0.238836\pi$
$379$			26.5494i			1.36375i	−0.731467	+	0.681877i	$0.761164\pi$
$380$	11.8968			0.610290
$381$		0			0
$382$		−	0.263254i		−	0.0134693i
$383$			3.27501i			0.167345i	0.996493	+	0.0836727i	$0.0266650\pi$
$383$			3.27501i			0.167345i	−0.996493	+	0.0836727i	$0.973335\pi$
$384$		0			0
$385$		−	18.7364i		−	0.954895i
$386$	11.7167			0.596367
$387$		0			0
$388$		−	0.618592i		−	0.0314042i
$389$	25.5198			1.29390			0.646951	−	0.762531i	$-0.276044\pi$
$389$	25.5198			1.29390			0.646951	+	0.762531i	$0.276044\pi$
$390$		0			0
$391$	26.1250			1.32120
$392$			4.51964i			0.228276i
$393$		0			0
$394$	−20.5283			−1.03420
$395$			40.0521i			2.01524i
$396$		0			0
$397$			4.70957i			0.236367i	0.992992	+	0.118183i	$0.0377071\pi$
$397$			4.70957i			0.236367i	−0.992992	+	0.118183i	$0.962293\pi$
$398$		−	5.95247i		−	0.298371i
$399$		0			0
$400$	−4.97624			−0.248812
$401$		−	14.4833i		−	0.723263i	−0.932321	−	0.361631i	$-0.882220\pi$
$401$		−	14.4833i		−	0.723263i	0.932321	−	0.361631i	$-0.117780\pi$
$402$		0			0
$403$	−7.74865	+	7.90002i	−0.385988	+	0.393528i
$404$	−8.33563			−0.414713
$405$		0			0
$406$	−0.343734			−0.0170592
$407$	−1.10031			−0.0545403
$408$		0			0
$409$			16.4189i			0.811860i	0.913904	+	0.405930i	$0.133052\pi$
$409$			16.4189i			0.811860i	−0.913904	+	0.405930i	$0.866948\pi$
$410$			23.3152i			1.15145i
$411$		0			0
$412$	9.89675			0.487578
$413$	−16.4533			−0.809614
$414$		0			0
$415$	1.14810			0.0563582
$416$	2.52473	−	2.57405i	0.123785	−	0.126203i
$417$		0			0
$418$			14.1870i			0.693908i
$419$	14.3082			0.699002			0.349501	−	0.936936i	$-0.386351\pi$
$419$	14.3082			0.699002			0.349501	+	0.936936i	$0.386351\pi$
$420$		0			0
$421$			24.0949i			1.17432i	0.809472	+	0.587158i	$0.199753\pi$
$421$			24.0949i			1.17432i	−0.809472	+	0.587158i	$0.800247\pi$
$422$			17.3925i			0.846653i
$423$		0			0
$424$		−	14.4175i		−	0.700178i
$425$	−35.1604			−1.70553
$426$		0			0
$427$		−	9.44971i		−	0.457304i
$428$	−0.115559			−0.00558574
$429$		0			0
$430$	−19.3197			−0.931680
$431$			9.12376i			0.439476i	0.975559	+	0.219738i	$0.0705202\pi$
$431$			9.12376i			0.439476i	−0.975559	+	0.219738i	$0.929480\pi$
$432$		0			0
$433$	0.372606			0.0179063			0.00895315	−	0.999960i	$-0.497150\pi$
$433$	0.372606			0.0179063			0.00895315	+	0.999960i	$0.497150\pi$
$434$			4.83358i			0.232019i
$435$		0			0
$436$		−	14.1301i		−	0.676708i
$437$		−	13.9267i		−	0.666205i
$438$		0			0
$439$	−3.82319			−0.182471			−0.0912355	−	0.995829i	$-0.529082\pi$
$439$	−3.82319			−0.182471			−0.0912355	+	0.995829i	$0.529082\pi$
$440$		−	11.8968i		−	0.567155i
$441$		0			0
$442$	17.8389	−	18.1874i	0.848508	−	0.865085i
$443$	−7.96406			−0.378384			−0.189192	−	0.981940i	$-0.560587\pi$
$443$	−7.96406			−0.378384			−0.189192	+	0.981940i	$0.560587\pi$
$444$		0			0
$445$	22.3167			1.05791
$446$	23.7456			1.12439
$447$		0			0
$448$		−	1.57492i		−	0.0744078i
$449$		−	26.4088i		−	1.24631i	−0.782098	−	0.623155i	$-0.785851\pi$
$449$		−	26.4088i		−	1.24631i	0.782098	−	0.623155i	$-0.214149\pi$
$450$		0			0
$451$	−27.8035			−1.30922
$452$	−19.4194			−0.913410
$453$		0			0
$454$	4.54135			0.213136
$455$	12.8044	+	12.5590i	0.600278	+	0.588776i
$456$		0			0
$457$			8.61483i			0.402985i	0.979490	+	0.201492i	$0.0645791\pi$
$457$			8.61483i			0.402985i	−0.979490	+	0.201492i	$0.935421\pi$
$458$	−11.9711			−0.559374
$459$		0			0
$460$			11.6785i			0.544513i
$461$		−	27.7998i		−	1.29477i	−0.762164	−	0.647384i	$-0.775863\pi$
$461$		−	27.7998i		−	1.29477i	0.762164	−	0.647384i	$-0.224137\pi$
$462$		0			0
$463$			36.3097i			1.68745i	0.536773	+	0.843727i	$0.319643\pi$
$463$			36.3097i			1.68745i	−0.536773	+	0.843727i	$0.680357\pi$
$464$	−0.218255			−0.0101323
$465$		0			0
$466$		−	10.6895i		−	0.495180i
$467$	13.8295			0.639954			0.319977	−	0.947425i	$-0.396325\pi$
$467$	13.8295			0.639954			0.319977	+	0.947425i	$0.396325\pi$
$468$		0			0
$469$	11.5221			0.532040
$470$			22.5353i			1.03947i
$471$		0			0
$472$	−10.4471			−0.480866
$473$		−	23.0389i		−	1.05933i
$474$		0			0
$475$			18.7433i			0.860002i
$476$		−	11.1278i		−	0.510043i
$477$		0			0
$478$	−24.0968			−1.10216
$479$			28.2335i			1.29002i	0.764174	+	0.645010i	$0.223147\pi$
$479$			28.2335i			1.29002i	−0.764174	+	0.645010i	$0.776853\pi$
$480$		0			0
$481$	0.737538	−	0.751946i	0.0336288	−	0.0342858i
$482$	11.1299			0.506954
$483$		0			0
$484$	3.18698			0.144863
$485$	1.95383			0.0887190
$486$		0			0
$487$		−	15.5579i		−	0.704995i	−0.935813	−	0.352498i	$-0.885332\pi$
$487$		−	15.5579i		−	0.704995i	0.935813	−	0.352498i	$-0.114668\pi$
$488$		−	6.00013i		−	0.271613i
$489$		0			0
$490$	−14.2754			−0.644895
$491$	14.9787			0.675980			0.337990	−	0.941150i	$-0.390253\pi$
$491$	14.9787			0.675980			0.337990	+	0.941150i	$0.390253\pi$
$492$		0			0
$493$	−1.54212			−0.0694535
$494$	−9.69532	−	9.50955i	−0.436213	−	0.427855i
$495$		0			0
$496$			3.06910i			0.137807i
$497$	1.21648			0.0545666
$498$		0			0
$499$			34.1030i			1.52666i	0.646008	+	0.763330i	$0.276437\pi$
$499$			34.1030i			1.52666i	−0.646008	+	0.763330i	$0.723563\pi$
$500$			0.0750557i			0.00335659i
$501$		0			0
$502$		−	11.3356i		−	0.505934i
$503$	−22.7259			−1.01330			−0.506648	−	0.862153i	$-0.669116\pi$
$503$	−22.7259			−1.01330			−0.506648	+	0.862153i	$0.669116\pi$
$504$		0			0
$505$		−	26.3282i		−	1.17159i
$506$	−13.9267			−0.619118
$507$		0			0
$508$	−5.42904			−0.240875
$509$			6.33426i			0.280761i	0.990098	+	0.140380i	$0.0448326\pi$
$509$			6.33426i			0.280761i	−0.990098	+	0.140380i	$0.955167\pi$
$510$		0			0
$511$	21.3153			0.942933
$512$		−	1.00000i		−	0.0441942i
$513$		0			0
$514$		−	8.32053i		−	0.367003i
$515$			31.2591i			1.37744i
$516$		0			0
$517$	−26.8735			−1.18189
$518$		−	0.460073i		−	0.0202145i
$519$		0			0
$520$	8.13019	+	7.97440i	0.356532	+	0.349701i
$521$	19.5169			0.855052			0.427526	−	0.904003i	$-0.359385\pi$
$521$	19.5169			0.855052			0.427526	+	0.904003i	$0.359385\pi$
$522$		0			0
$523$	4.76549			0.208380			0.104190	−	0.994557i	$-0.466775\pi$
$523$	4.76549			0.208380			0.104190	+	0.994557i	$0.466775\pi$
$524$	12.6868			0.554227
$525$		0			0
$526$			14.3827i			0.627113i
$527$			21.6852i			0.944622i
$528$		0			0
$529$	−9.32879			−0.405599
$530$	45.5381			1.97805
$531$		0			0
$532$	−5.93202			−0.257186
$533$	18.6367	−	19.0008i	0.807247	−	0.823017i
$534$		0			0
$535$		−	0.364994i		−	0.0157801i
$536$	7.31599			0.316002
$537$		0			0
$538$		−	9.01366i		−	0.388606i
$539$		−	17.0235i		−	0.733254i
$540$		0			0
$541$			25.2325i			1.08483i	0.840111	+	0.542414i	$0.182490\pi$
$541$			25.2325i			1.08483i	−0.840111	+	0.542414i	$0.817510\pi$
$542$	19.9094			0.855182
$543$		0			0
$544$		−	7.06565i		−	0.302937i
$545$	44.6301			1.91174
$546$		0			0
$547$	28.7833			1.23069			0.615343	−	0.788260i	$-0.289018\pi$
$547$	28.7833			1.23069			0.615343	+	0.788260i	$0.289018\pi$
$548$		−	7.35541i		−	0.314207i
$549$		0			0
$550$	18.7433			0.799217
$551$			0.822072i			0.0350215i
$552$		0			0
$553$		−	19.9710i		−	0.849253i
$554$		−	13.3831i		−	0.568595i
$555$		0			0
$556$	7.31119			0.310063
$557$		−	11.4847i		−	0.486624i	−0.969948	−	0.243312i	$-0.921766\pi$
$557$		−	11.4847i		−	0.486624i	0.969948	−	0.243312i	$-0.0782339\pi$
$558$		0			0
$559$	15.7447	+	15.4430i	0.665931	+	0.653170i
$560$	4.97440			0.210207
$561$		0			0
$562$	0.700291			0.0295400
$563$	14.8478			0.625759			0.312879	−	0.949793i	$-0.398706\pi$
$563$	14.8478			0.625759			0.312879	+	0.949793i	$0.398706\pi$
$564$		0			0
$565$		−	61.3364i		−	2.58044i
$566$		−	19.4357i		−	0.816942i
$567$		0			0
$568$	0.772410			0.0324096
$569$	27.7762			1.16444			0.582220	−	0.813031i	$-0.302184\pi$
$569$	27.7762			1.16444			0.582220	+	0.813031i	$0.302184\pi$
$570$		0			0
$571$	−19.1208			−0.800180			−0.400090	−	0.916476i	$-0.631021\pi$
$571$	−19.1208			−0.800180			−0.400090	+	0.916476i	$0.631021\pi$
$572$	−9.50955	+	9.69532i	−0.397614	+	0.405382i
$573$		0			0
$574$		−	11.6255i		−	0.485240i
$575$	−18.3994			−0.767310
$576$		0			0
$577$			26.8961i			1.11970i	0.828595	+	0.559849i	$0.189141\pi$
$577$			26.8961i			1.11970i	−0.828595	+	0.559849i	$0.810859\pi$
$578$		−	32.9235i		−	1.36944i
$579$		0			0
$580$		−	0.689364i		−	0.0286243i
$581$	−0.572474			−0.0237502
$582$		0			0
$583$			54.3046i			2.24907i
$584$	13.5342			0.560051
$585$		0			0
$586$	4.25760			0.175880
$587$		−	39.0047i		−	1.60990i	−0.593344	−	0.804949i	$-0.702193\pi$
$587$		−	39.0047i		−	1.60990i	0.593344	−	0.804949i	$-0.297807\pi$
$588$		0			0
$589$	11.5600			0.476320
$590$		−	32.9973i		−	1.35848i
$591$		0			0
$592$		−	0.292126i		−	0.0120063i
$593$		−	24.2756i		−	0.996881i	−0.866924	−	0.498441i	$-0.833906\pi$
$593$		−	24.2756i		−	0.996881i	0.866924	−	0.498441i	$-0.166094\pi$
$594$		0			0
$595$	35.1474			1.44090
$596$			23.7638i			0.973401i
$597$		0			0
$598$	9.33509	−	9.51746i	0.381740	−	0.389198i
$599$	−19.0750			−0.779385			−0.389692	−	0.920945i	$-0.627419\pi$
$599$	−19.0750			−0.779385			−0.389692	+	0.920945i	$0.627419\pi$
$600$		0			0
$601$	−14.8668			−0.606431			−0.303216	−	0.952922i	$-0.598060\pi$
$601$	−14.8668			−0.606431			−0.303216	+	0.952922i	$0.598060\pi$
$602$	9.63330			0.392624
$603$		0			0
$604$			10.3890i			0.422721i
$605$			10.0661i			0.409247i
$606$		0			0
$607$	36.7796			1.49284			0.746419	−	0.665476i	$-0.231771\pi$
$607$	36.7796			1.49284			0.746419	+	0.665476i	$0.231771\pi$
$608$	−3.76656			−0.152754
$609$		0			0
$610$	18.9515			0.767325
$611$	18.0133	−	18.3652i	0.728741	−	0.742978i
$612$		0			0
$613$			9.13501i			0.368959i	0.982836	+	0.184480i	$0.0590600\pi$
$613$			9.13501i			0.368959i	−0.982836	+	0.184480i	$0.940940\pi$
$614$	−7.35088			−0.296657
$615$		0			0
$616$			5.93202i			0.239008i
$617$		−	14.2610i		−	0.574127i	−0.957912	−	0.287063i	$-0.907321\pi$
$617$		−	14.2610i		−	0.574127i	0.957912	−	0.287063i	$-0.0926791\pi$
$618$		0			0
$619$		−	5.96528i		−	0.239765i	−0.992788	−	0.119883i	$-0.961748\pi$
$619$		−	5.96528i		−	0.239765i	0.992788	−	0.119883i	$-0.0382518\pi$
$620$	−9.69381			−0.389313
$621$		0			0
$622$			2.81511i			0.112876i
$623$	−11.1276			−0.445820
$624$		0			0
$625$	−25.1182			−1.00473
$626$		−	27.0917i		−	1.08280i
$627$		0			0
$628$	9.14824			0.365054
$629$		−	2.06406i		−	0.0822994i
$630$		0			0
$631$		−	3.19802i		−	0.127311i	−0.997972	−	0.0636557i	$-0.979724\pi$
$631$		−	3.19802i		−	0.127311i	0.997972	−	0.0636557i	$-0.0202759\pi$
$632$		−	12.6807i		−	0.504410i
$633$		0			0
$634$	0.756736			0.0300538
$635$		−	17.1477i		−	0.680487i
$636$		0			0
$637$	11.6338	+	11.4109i	0.460947	+	0.452115i
$638$	0.822072			0.0325462
$639$		0			0
$640$	3.15852			0.124851
$641$	30.9602			1.22285			0.611426	−	0.791301i	$-0.290596\pi$
$641$	30.9602			1.22285			0.611426	+	0.791301i	$0.290596\pi$
$642$		0			0
$643$		−	29.4516i		−	1.16146i	−0.814097	−	0.580729i	$-0.802768\pi$
$643$		−	29.4516i		−	1.16146i	0.814097	−	0.580729i	$-0.197232\pi$
$644$		−	5.82319i		−	0.229466i
$645$		0			0
$646$	−26.6132			−1.04708
$647$	−27.8656			−1.09551			−0.547756	−	0.836638i	$-0.684518\pi$
$647$	−27.8656			−1.09551			−0.547756	+	0.836638i	$0.684518\pi$
$648$		0			0
$649$	39.3496			1.54461
$650$	−12.5636	+	12.8091i	−0.492787	+	0.502414i
$651$		0			0
$652$			13.5935i			0.532363i
$653$	18.9723			0.742445			0.371223	−	0.928544i	$-0.378939\pi$
$653$	18.9723			0.742445			0.371223	+	0.928544i	$0.378939\pi$
$654$		0			0
$655$			40.0716i			1.56573i
$656$		−	7.38168i		−	0.288206i
$657$		0			0
$658$		−	11.2366i		−	0.438050i
$659$	18.1177			0.705766			0.352883	−	0.935667i	$-0.385201\pi$
$659$	18.1177			0.705766			0.352883	+	0.935667i	$0.385201\pi$
$660$		0			0
$661$		−	23.7528i		−	0.923878i	−0.886912	−	0.461939i	$-0.847154\pi$
$661$		−	23.7528i		−	0.923878i	0.886912	−	0.461939i	$-0.152846\pi$
$662$	−32.9672			−1.28131
$663$		0			0
$664$	−0.363495			−0.0141063
$665$		−	18.7364i		−	0.726566i
$666$		0			0
$667$	−0.806991			−0.0312468
$668$		−	7.16615i		−	0.277267i
$669$		0			0
$670$			23.1077i			0.892727i
$671$			22.5999i			0.872458i
$672$		0			0
$673$	24.2314			0.934052			0.467026	−	0.884244i	$-0.345325\pi$
$673$	24.2314			0.934052			0.467026	+	0.884244i	$0.345325\pi$
$674$		−	4.50695i		−	0.173601i
$675$		0			0
$676$	−0.251488	−	12.9976i	−0.00967261	−	0.499906i
$677$	19.5400			0.750985			0.375492	−	0.926826i	$-0.377474\pi$
$677$	19.5400			0.750985			0.375492	+	0.926826i	$0.377474\pi$
$678$		0			0
$679$	−0.974230			−0.0373875
$680$	22.3170			0.855818
$681$		0			0
$682$		−	11.5600i		−	0.442654i
$683$			40.5718i			1.55244i	0.630465	+	0.776218i	$0.282864\pi$
$683$			40.5718i			1.55244i	−0.630465	+	0.776218i	$0.717136\pi$
$684$		0			0
$685$	23.2322			0.887656
$686$	18.1425			0.692683
$687$		0			0
$688$	6.11671			0.233197
$689$	−37.1115	−	36.4004i	−1.41384	−	1.38674i
$690$		0			0
$691$			29.6026i			1.12614i	0.826410	+	0.563069i	$0.190379\pi$
$691$			29.6026i			1.12614i	−0.826410	+	0.563069i	$0.809621\pi$
$692$	−3.01163			−0.114485
$693$		0			0
$694$			29.9717i			1.13771i
$695$			23.0925i			0.875949i
$696$		0			0
$697$		−	52.1564i		−	1.97556i
$698$	17.4351			0.659927
$699$		0			0
$700$			7.83716i			0.296217i
$701$	−14.6213			−0.552238			−0.276119	−	0.961123i	$-0.589048\pi$
$701$	−14.6213			−0.552238			−0.276119	+	0.961123i	$0.589048\pi$
$702$		0			0
$703$	−1.10031			−0.0414989
$704$			3.76656i			0.141958i
$705$		0			0
$706$	4.51846			0.170054
$707$			13.1279i			0.493726i
$708$		0			0
$709$		−	6.39968i		−	0.240345i	−0.992753	−	0.120173i	$-0.961655\pi$
$709$		−	6.39968i		−	0.240345i	0.992753	−	0.120173i	$-0.0383448\pi$
$710$			2.43967i			0.0915592i
$711$		0			0
$712$	−7.06555			−0.264793
$713$			11.3479i			0.424981i
$714$		0			0
$715$	−30.6229	−	30.0361i	−1.14523	−	1.12329i
$716$	19.2107			0.717936
$717$		0			0
$718$	30.5681			1.14079
$719$	13.5954			0.507022			0.253511	−	0.967332i	$-0.418415\pi$
$719$	13.5954			0.507022			0.253511	+	0.967332i	$0.418415\pi$
$720$		0			0
$721$		−	15.5866i		−	0.580474i
$722$		−	4.81302i		−	0.179122i
$723$		0			0
$724$	−9.14810			−0.339987
$725$	1.08609			0.0403364
$726$		0			0
$727$	−22.2352			−0.824657			−0.412329	−	0.911035i	$-0.635284\pi$
$727$	−22.2352			−0.824657			−0.412329	+	0.911035i	$0.635284\pi$
$728$	−4.05392	−	3.97624i	−0.150248	−	0.147369i
$729$		0			0
$730$			42.7481i			1.58218i
$731$	43.2185			1.59850
$732$		0			0
$733$			17.0624i			0.630213i	0.949056	+	0.315107i	$0.102040\pi$
$733$			17.0624i			0.630213i	−0.949056	+	0.315107i	$0.897960\pi$
$734$			10.6684i			0.393778i
$735$		0			0
$736$		−	3.69746i		−	0.136290i
$737$	−27.5561			−1.01504
$738$		0			0
$739$			47.4169i			1.74426i	0.489274	+	0.872130i	$0.337262\pi$
$739$			47.4169i			1.74426i	−0.489274	+	0.872130i	$0.662738\pi$
$740$	0.922684			0.0339185
$741$		0			0
$742$	−22.7064			−0.833579
$743$			26.2945i			0.964650i	0.875992	+	0.482325i	$0.160208\pi$
$743$			26.2945i			0.964650i	−0.875992	+	0.482325i	$0.839792\pi$
$744$		0			0
$745$	−75.0583			−2.74992
$746$		−	8.36446i		−	0.306245i
$747$		0			0
$748$			26.6132i			0.973076i
$749$			0.181995i			0.00664997i
$750$		0			0
$751$	33.8510			1.23524			0.617620	−	0.786477i	$-0.288097\pi$
$751$	33.8510			1.23524			0.617620	+	0.786477i	$0.288097\pi$
$752$		−	7.13476i		−	0.260178i
$753$		0			0
$754$	−0.551036	+	0.561801i	−0.0200675	+	0.0204596i
$755$	−32.8137			−1.19421
$756$		0			0
$757$	−35.4490			−1.28841			−0.644207	−	0.764851i	$-0.722813\pi$
$757$	−35.4490			−1.28841			−0.644207	+	0.764851i	$0.722813\pi$
$758$	26.5494			0.964319
$759$		0			0
$760$		−	11.8968i		−	0.431541i
$761$			27.3763i			0.992391i	0.868211	+	0.496196i	$0.165270\pi$
$761$			27.3763i			0.992391i	−0.868211	+	0.496196i	$0.834730\pi$
$762$		0			0
$763$	−22.2537			−0.805638
$764$	−0.263254			−0.00952420
$765$		0			0
$766$	3.27501			0.118331
$767$	−26.3761	+	26.8914i	−0.952385	+	0.970991i
$768$		0			0
$769$		−	14.8890i		−	0.536911i	−0.963292	−	0.268456i	$-0.913487\pi$
$769$		−	14.8890i		−	0.536911i	0.963292	−	0.268456i	$-0.0865133\pi$
$770$	−18.7364			−0.675213
$771$		0			0
$772$		−	11.7167i		−	0.421695i
$773$			47.2807i			1.70057i	0.526324	+	0.850284i	$0.323570\pi$
$773$			47.2807i			1.70057i	−0.526324	+	0.850284i	$0.676430\pi$
$774$		0			0
$775$		−	15.2726i		−	0.548607i
$776$	−0.618592			−0.0222061
$777$		0			0
$778$		−	25.5198i		−	0.914927i
$779$	−27.8035			−0.996165
$780$		0			0
$781$	−2.90933			−0.104104
$782$		−	26.1250i		−	0.934227i
$783$		0			0
$784$	4.51964			0.161416
$785$			28.8949i			1.03130i
$786$		0			0
$787$			1.42184i			0.0506832i	0.999679	+	0.0253416i	$0.00806735\pi$
$787$			1.42184i			0.0506832i	−0.999679	+	0.0253416i	$0.991933\pi$
$788$			20.5283i			0.731291i
$789$		0			0
$790$	40.0521			1.42499
$791$			30.5839i			1.08744i
$792$		0			0
$793$	−15.4447	−	15.1487i	−0.548456	−	0.537947i
$794$	4.70957			0.167137
$795$		0			0
$796$	−5.95247			−0.210980
$797$	−43.6946			−1.54774			−0.773872	−	0.633342i	$-0.781683\pi$
$797$	−43.6946			−1.54774			−0.773872	+	0.633342i	$0.781683\pi$
$798$		0			0
$799$		−	50.4117i		−	1.78344i
$800$			4.97624i			0.175937i
$801$		0			0
$802$	−14.4833			−0.511424
$803$	−50.9775			−1.79896
$804$		0			0
$805$	18.3927			0.648256
$806$	7.90002	+	7.74865i	0.278266	+	0.272934i
$807$		0			0
$808$			8.33563i			0.293246i
$809$	5.70765			0.200670			0.100335	−	0.994954i	$-0.468008\pi$
$809$	5.70765			0.200670			0.100335	+	0.994954i	$0.468008\pi$
$810$		0			0
$811$		−	50.0141i		−	1.75623i	−0.478446	−	0.878117i	$-0.658800\pi$
$811$		−	50.0141i		−	1.75623i	0.478446	−	0.878117i	$-0.341200\pi$
$812$			0.343734i			0.0120627i
$813$		0			0
$814$			1.10031i			0.0385658i
$815$	−42.9354			−1.50396
$816$		0			0
$817$		−	23.0389i		−	0.806031i
$818$	16.4189			0.574072
$819$		0			0
$820$	23.3152			0.814201
$821$		−	52.1953i		−	1.82163i	−0.412815	−	0.910815i	$-0.635454\pi$
$821$		−	52.1953i		−	1.82163i	0.412815	−	0.910815i	$-0.364546\pi$
$822$		0			0
$823$	−37.0555			−1.29167			−0.645837	−	0.763475i	$-0.723491\pi$
$823$	−37.0555			−1.29167			−0.645837	+	0.763475i	$0.723491\pi$
$824$		−	9.89675i		−	0.344770i
$825$		0			0
$826$			16.4533i			0.572483i
$827$			14.2975i			0.497174i	0.968610	+	0.248587i	$0.0799662\pi$
$827$			14.2975i			0.497174i	−0.968610	+	0.248587i	$0.920034\pi$
$828$		0			0
$829$	−29.4062			−1.02132			−0.510659	−	0.859783i	$-0.670599\pi$
$829$	−29.4062			−1.02132			−0.510659	+	0.859783i	$0.670599\pi$
$830$		−	1.14810i		−	0.0398513i
$831$		0			0
$832$	−2.57405	−	2.52473i	−0.0892392	−	0.0875292i
$833$	31.9342			1.10645
$834$		0			0
$835$	22.6344			0.783297
$836$	14.1870			0.490667
$837$		0			0
$838$		−	14.3082i		−	0.494269i
$839$			12.1057i			0.417934i	0.977923	+	0.208967i	$0.0670102\pi$
$839$			12.1057i			0.417934i	−0.977923	+	0.208967i	$0.932990\pi$
$840$		0			0
$841$	−28.9524			−0.998357
$842$	24.0949			0.830367
$843$		0			0
$844$	17.3925			0.598674
$845$	41.0531	−	0.794329i	1.41227	−	0.0273257i
$846$		0			0
$847$		−	5.01923i		−	0.172463i
$848$	−14.4175			−0.495101
$849$		0			0
$850$			35.1604i			1.20599i
$851$		−	1.08012i		−	0.0370261i
$852$		0			0
$853$		−	0.747177i		−	0.0255828i	−0.999918	−	0.0127914i	$-0.995928\pi$
$853$		−	0.747177i		−	0.0255828i	0.999918	−	0.0127914i	$-0.00407175\pi$
$854$	−9.44971			−0.323362
$855$		0			0
$856$			0.115559i			0.00394972i
$857$	−50.0722			−1.71043			−0.855217	−	0.518269i	$-0.826576\pi$
$857$	−50.0722			−1.71043			−0.855217	+	0.518269i	$0.826576\pi$
$858$		0			0
$859$	28.7360			0.980460			0.490230	−	0.871593i	$-0.336913\pi$
$859$	28.7360			0.980460			0.490230	+	0.871593i	$0.336913\pi$
$860$			19.3197i			0.658797i
$861$		0			0
$862$	9.12376			0.310756
$863$		−	6.03591i		−	0.205465i	−0.994709	−	0.102732i	$-0.967241\pi$
$863$		−	6.03591i		−	0.205465i	0.994709	−	0.102732i	$-0.0327585\pi$
$864$		0			0
$865$		−	9.51229i		−	0.323428i
$866$		−	0.372606i		−	0.0126617i
$867$		0			0
$868$	4.83358			0.164062
$869$			47.7625i			1.62023i
$870$		0			0
$871$	18.4709	−	18.8317i	0.625862	−	0.638089i
$872$	−14.1301			−0.478505
$873$		0			0
$874$	−13.9267			−0.471078
$875$	0.118206			0.00399611
$876$		0			0
$877$			15.9617i			0.538990i	0.963002	+	0.269495i	$0.0868567\pi$
$877$			15.9617i			0.538990i	−0.963002	+	0.269495i	$0.913143\pi$
$878$			3.82319i			0.129026i
$879$		0			0
$880$	−11.8968			−0.401039
$881$	−18.2622			−0.615269			−0.307635	−	0.951505i	$-0.599537\pi$
$881$	−18.2622			−0.615269			−0.307635	+	0.951505i	$0.599537\pi$
$882$		0			0
$883$	6.17305			0.207740			0.103870	−	0.994591i	$-0.466877\pi$
$883$	6.17305			0.207740			0.103870	+	0.994591i	$0.466877\pi$
$884$	−18.1874	−	17.8389i	−0.611707	−	0.599986i
$885$		0			0
$886$			7.96406i			0.267558i
$887$	10.0193			0.336414			0.168207	−	0.985752i	$-0.446202\pi$
$887$	10.0193			0.336414			0.168207	+	0.985752i	$0.446202\pi$
$888$		0			0
$889$			8.55029i			0.286767i
$890$		−	22.3167i		−	0.748056i
$891$		0			0
$892$		−	23.7456i		−	0.795062i
$893$	−26.8735			−0.899287
$894$		0			0
$895$			60.6772i			2.02821i
$896$	−1.57492			−0.0526143
$897$		0			0
$898$	−26.4088			−0.881274
$899$		−	0.669848i		−	0.0223407i
$900$		0			0
$901$	−101.869			−3.39376
$902$			27.8035i			0.925757i
$903$		0			0
$904$			19.4194i			0.645878i
$905$		−	28.8945i		−	0.960484i
$906$		0			0
$907$	−57.5470			−1.91082			−0.955408	−	0.295290i	$-0.904584\pi$
$907$	−57.5470			−1.91082			−0.955408	+	0.295290i	$0.904584\pi$
$908$		−	4.54135i		−	0.150710i
$909$		0			0
$910$	12.5590	−	12.8044i	0.416327	−	0.424461i
$911$	−15.7798			−0.522808			−0.261404	−	0.965229i	$-0.584186\pi$
$911$	−15.7798			−0.522808			−0.261404	+	0.965229i	$0.584186\pi$
$912$		0			0
$913$	1.36912			0.0453114
$914$	8.61483			0.284953
$915$		0			0
$916$			11.9711i			0.395537i
$917$		−	19.9807i		−	0.659821i
$918$		0			0
$919$	42.9349			1.41629			0.708146	−	0.706066i	$-0.249532\pi$
$919$	42.9349			1.41629			0.708146	+	0.706066i	$0.249532\pi$
$920$	11.6785			0.385029
$921$		0			0
$922$	−27.7998			−0.915539
$923$	1.95013	−	1.98822i	0.0641892	−	0.0654432i
$924$		0			0
$925$			1.45369i			0.0477969i
$926$	36.3097			1.19321
$927$		0			0
$928$			0.218255i			0.00716458i
$929$		−	10.3748i		−	0.340387i	−0.985411	−	0.170193i	$-0.945561\pi$
$929$		−	10.3748i		−	0.340387i	0.985411	−	0.170193i	$-0.0544392\pi$
$930$		0			0
$931$		−	17.0235i		−	0.557922i
$932$	−10.6895			−0.350145
$933$		0			0
$934$		−	13.8295i		−	0.452516i
$935$	−84.0583			−2.74900
$936$		0			0
$937$	−26.4110			−0.862809			−0.431404	−	0.902159i	$-0.641982\pi$
$937$	−26.4110			−0.862809			−0.431404	+	0.902159i	$0.641982\pi$
$938$		−	11.5221i		−	0.376209i
$939$		0			0
$940$	22.5353			0.735019
$941$			12.7869i			0.416842i	0.978039	+	0.208421i	$0.0668324\pi$
$941$			12.7869i			0.416842i	−0.978039	+	0.208421i	$0.933168\pi$
$942$		0			0
$943$		−	27.2935i		−	0.888797i
$944$			10.4471i			0.340024i
$945$		0			0
$946$	−23.0389			−0.749061
$947$		−	0.462902i		−	0.0150423i	−0.999972	−	0.00752114i	$-0.997606\pi$
$947$		−	0.462902i		−	0.0150423i	0.999972	−	0.00752114i	$-0.00239408\pi$
$948$		0			0
$949$	34.1703	−	34.8378i	1.10921	−	1.13088i
$950$	18.7433			0.608113
$951$		0			0
$952$	−11.1278			−0.360655
$953$	12.0901			0.391636			0.195818	−	0.980640i	$-0.437264\pi$
$953$	12.0901			0.391636			0.195818	+	0.980640i	$0.437264\pi$
$954$		0			0
$955$		−	0.831493i		−	0.0269065i
$956$			24.0968i			0.779346i
$957$		0			0
$958$	28.2335			0.912182
$959$	−11.5841			−0.374072
$960$		0			0
$961$	21.5806			0.696149
$962$	−0.751946	−	0.737538i	−0.0242437	−	0.0237792i
$963$		0			0
$964$		−	11.1299i		−	0.358470i
$965$	37.0076			1.19132
$966$		0			0
$967$		−	29.8157i		−	0.958808i	−0.877594	−	0.479404i	$-0.840853\pi$
$967$		−	29.8157i		−	0.958808i	0.877594	−	0.479404i	$-0.159147\pi$
$968$		−	3.18698i		−	0.102433i
$969$		0			0
$970$		−	1.95383i		−	0.0627338i
$971$	−61.4263			−1.97126			−0.985632	−	0.168908i	$-0.945976\pi$
$971$	−61.4263			−1.97126			−0.985632	+	0.168908i	$0.945976\pi$
$972$		0			0
$973$		−	11.5145i		−	0.369138i
$974$	−15.5579			−0.498507
$975$		0			0
$976$	−6.00013			−0.192060
$977$			22.8673i			0.731589i	0.930696	+	0.365794i	$0.119203\pi$
$977$			22.8673i			0.731589i	−0.930696	+	0.365794i	$0.880797\pi$
$978$		0			0
$979$	26.6128			0.850549
$980$			14.2754i			0.456010i
$981$		0			0
$982$		−	14.9787i		−	0.477990i
$983$		−	28.9097i		−	0.922076i	−0.887380	−	0.461038i	$-0.847477\pi$
$983$		−	28.9097i		−	0.922076i	0.887380	−	0.461038i	$-0.152523\pi$
$984$		0			0
$985$	−64.8391			−2.06594
$986$			1.54212i			0.0491110i
$987$		0			0
$988$	−9.50955	+	9.69532i	−0.302539	+	0.308449i
$989$	22.6163			0.719156
$990$		0			0
$991$	21.3095			0.676917			0.338459	−	0.940981i	$-0.390095\pi$
$991$	21.3095			0.676917			0.338459	+	0.940981i	$0.390095\pi$
$992$	3.06910			0.0974440
$993$		0			0
$994$		−	1.21648i		−	0.0385844i
$995$		−	18.8010i		−	0.596032i
$996$		0			0
$997$	−53.8074			−1.70410			−0.852049	−	0.523462i	$-0.824640\pi$
$997$	−53.8074			−1.70410			−0.852049	+	0.523462i	$0.824640\pi$
$998$	34.1030			1.07951
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2106.2.b.c.649.1		14
3.2	odd	2		2106.2.b.d.649.14		14
9.2	odd	6		702.2.t.a.415.8		28
9.4	even	3		234.2.t.a.25.13	yes	28
9.5	odd	6		702.2.t.a.181.7		28
9.7	even	3		234.2.t.a.103.6	yes	28
13.12	even	2	inner	2106.2.b.c.649.14		14
39.38	odd	2		2106.2.b.d.649.1		14
117.25	even	6		234.2.t.a.103.13	yes	28
117.38	odd	6		702.2.t.a.415.7		28
117.77	odd	6		702.2.t.a.181.8		28
117.103	even	6		234.2.t.a.25.6	✓	28

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
234.2.t.a.25.6	✓	28	117.103	even	6
234.2.t.a.25.13	yes	28	9.4	even	3
234.2.t.a.103.6	yes	28	9.7	even	3
234.2.t.a.103.13	yes	28	117.25	even	6
702.2.t.a.181.7		28	9.5	odd	6
702.2.t.a.181.8		28	117.77	odd	6
702.2.t.a.415.7		28	117.38	odd	6
702.2.t.a.415.8		28	9.2	odd	6
2106.2.b.c.649.1		14	1.1	even	1	trivial
2106.2.b.c.649.14		14	13.12	even	2	inner
2106.2.b.d.649.1		14	39.38	odd	2
2106.2.b.d.649.14		14	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2106 = 2 \cdot 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2106.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{2} -1.00000 q^{4} -3.15852i q^{5} +1.57492i q^{7} +1.00000i q^{8} -3.15852 q^{10} -3.76656i q^{11} +(2.57405 + 2.52473i) q^{13} +1.57492 q^{14} +1.00000 q^{16} +7.06565 q^{17} -3.76656i q^{19} +3.15852i q^{20} -3.76656 q^{22} +3.69746 q^{23} -4.97624 q^{25} +(2.52473 - 2.57405i) q^{26} -1.57492i q^{28} -0.218255 q^{29} +3.06910i q^{31} -1.00000i q^{32} -7.06565i q^{34} +4.97440 q^{35} -0.292126i q^{37} -3.76656 q^{38} +3.15852 q^{40} -7.38168i q^{41} +6.11671 q^{43} +3.76656i q^{44} -3.69746i q^{46} -7.13476i q^{47} +4.51964 q^{49} +4.97624i q^{50} +(-2.57405 - 2.52473i) q^{52} -14.4175 q^{53} -11.8968 q^{55} -1.57492 q^{56} +0.218255i q^{58} +10.4471i q^{59} -6.00013 q^{61} +3.06910 q^{62} -1.00000 q^{64} +(7.97440 - 8.13019i) q^{65} -7.31599i q^{67} -7.06565 q^{68} -4.97440i q^{70} -0.772410i q^{71} -13.5342i q^{73} -0.292126 q^{74} +3.76656i q^{76} +5.93202 q^{77} -12.6807 q^{79} -3.15852i q^{80} -7.38168 q^{82} +0.363495i q^{83} -22.3170i q^{85} -6.11671i q^{86} +3.76656 q^{88} +7.06555i q^{89} +(-3.97624 + 4.05392i) q^{91} -3.69746 q^{92} -7.13476 q^{94} -11.8968 q^{95} +0.618592i q^{97} -4.51964i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 14 q^{4} - 2 q^{13} - 8 q^{14} + 14 q^{16} + 8 q^{17} + 8 q^{23} - 14 q^{25} + 4 q^{26} + 16 q^{29} - 34 q^{35} + 4 q^{43} - 10 q^{49} + 2 q^{52} - 60 q^{53} + 8 q^{56} - 28 q^{61} + 34 q^{62} - 14 q^{64}+ \cdots - 8 q^{92}+O(q^{100}) \) 14 * q - 14 * q^4 - 2 * q^13 - 8 * q^14 + 14 * q^16 + 8 * q^17 + 8 * q^23 - 14 * q^25 + 4 * q^26 + 16 * q^29 - 34 * q^35 + 4 * q^43 - 10 * q^49 + 2 * q^52 - 60 * q^53 + 8 * q^56 - 28 * q^61 + 34 * q^62 - 14 * q^64 + 8 * q^65 - 8 * q^68 - 16 * q^74 + 24 * q^77 - 28 * q^79 - 24 * q^82 - 8 * q^92

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